Where each number below was read, and how it was checked.
-0.6589(3), AFQMC
Checked 2026-09-15 · arXiv PDF extracted locally with pypdf in layout mode, E rows of Table IV parsed by column position; no LLM transcription
Reported as. E = -42.17(2) (total), 8 x 8, U = 6, PBC-APBC
Table IV, "Total ground state energy (E) ... in the Hubbard model at half-filling, for U = 2, 4, 6, 8 ... for both PBC and PBC-APBC. Statistical errors are on the last digit." Row 8 x 8, column U = 6 PBC-APBC. Half filling: N_up = N_dn = 32. Cross-checks: the 8 x 8 PBC values at U = 4 and 8 agree with VarBench's own AFQMC exact rows to 1.3 sigma, and the 8 x 8, 10 x 10 and 12 x 12 PBC U = 8 values with the AFQMC references quoted by arXiv:2507.02644 (Table S1) to the printed digits.
-0.6574(2), VMC (HFDS)
Checked 2026-09-15 · PNAS supplementary PDF (via Europe PMC, PMC9371695) extracted locally, cross-read against the arXiv version; no LLM transcription
Reported as. -0.6574(2) per site, no variance
PNAS SI Table 5 (identical to arXiv SI Table V), "Variational energy per site in the L x L Hubbard model at half filling with periodic boundary conditions along one of the sides of the square and anti-periodic boundary conditions along the other side", row L = 8, column U = 6. SI Sec. 6: N_hidden = 16, alpha = 1 for 8x8, the settings in the VarBench method string. Column alignment checked as for square_36_PA_18_2. As a total, -42.0736 sits 2.3e-3 (relative) above the sign-free AFQMC value -42.17(2) for this lattice (Qin, Shi & Zhang, PRB 94, 085103 (2016), Table IV, PBC-APBC), as a variational energy must, and fits the paper's size trend at U = 6 (-0.68135, -0.6609, -0.6574 for L = 4, 6, 8).